Limit

Informally, $\lim_{x \to a} f(x) = L$ means: as $x$ gets arbitrarily close to $a$ (from either side), $f(x)$ gets arbitrarily close to $L$. The function does not have to be defined at $a$ itself, and even if it is, the function value $f(a)$ does not have to equal $L$.

The formal $\varepsilon$-$\delta$ definition demands: for every $\varepsilon > 0$ there exists $\delta > 0$ such that $|x - a| < \delta$ implies $|f(x) - L| < \varepsilon$.

Limits make precise the notion of "approaching but not equalling" — the engine behind derivatives ($h \to 0$) and integrals (Riemann sums with mesh $\to 0$). Many physical and economic models implicitly depend on limit reasoning.